698

DOC.

659 NOVEMBER

1918

space

(in

which

the

rigid body

would

then

have

to

be in

a

form

of

rest),

the

mentioned conditions

can

be realized.

If

this

is

not

satisfied,

I

do

not

see

how

the

“rigid body” ought

to

be defined

accordingly.-

Then

I

also have

another

question

on

my

mind with

regard

to

the

facts of

Weyl's

theory.[2]

Would it

not

be

possible

that,

instead

of

ƒ /Euvguvdxudxv,

the

“length” were given

by

an

expression

ƒ

/k

•

Euvguvdxudxv,

where

k means an

invariant function

of

x1, x2, x3,

x4

against

coordinate

transformations

expressible

by

guv

and

Qv,

which function

is

constant in

the

case

where

guv

=

6uv

and

which,

upon

introduction

of

Weyl’s

A-factor, changes

into const

.

A-1

.

k?

Then,

of

course,

all

length

relations would be invariants in

Weyl's sense,

and

we

would have

the

usual

length

measurement

for

Euclidean

metrics.[3]

Given,

for

inst.,

the

world

were

composed

in such

a

way

that

in it

a

point O

could be chosen

so

that the

geodesic

lines

starting

from

O

covered

the

world

com-

pletely

and

simply

(except

for

O

itself,

of

course),

then the

expression

e-

fpo

EvQvdxv

in which

the

integral

extends

over

the

geodesic

link between O and P,

would be

a

function

k

of

the

coordinates

of P, which has

the

required properties.[4]

Hilbert,

from whom

the

idea

of

the introduction

of

a k

factor

originates,

has

indicated the

equation

i/rE/'-i'

H

\

--

1

dn

k

dx.

=

0

to

determine

such

a

function[5] which,

interpreted

as a

differential equation for

the

k

to

be

determined,

is

invariant

against

coordinate

transformations

and,

in

addition,

remains

unchanged

if

guv

is

replaced by

A

•

guv,

Qv

by Qv+

1dA/Adxv,

and

k

by

A-1

•

k.

Perhaps through

invariant

constraints

a

solution

dependent

on

the

guv's

and

Qv's

could be chosen for

this

differential

equation

such

that

in

the

case

Qv

=

0

(v

=

1,

...,

4)

it

is

equal

to

the

constant

1

and,

upon

introduction

of

a

A-factor,

is

multiplied by

A-1.

I

would appreciate

hearing your opinion

on

these matters.-

Regarding

the

Nelson

topic,

I

thoroughly

understand

your

point

of

view.[6]

I

gladly

accept

the

prospect

of

discussing

it

with

you orally

when

the

occasion

presents

itself.-

Currently,

the

political

events

are

presumably

at

the

center

of

your

interest.

What

turn

destiny

will

take

“lies

in

the

lap

of

the

gods.”[7]

Most

cordial

regards,

yours,

Paul

Bernays.